The authors consider the model for financial market, in which \(n+1\) assets are traded. The price \(S_{t}^0\) of the first of these assets evolves according to the equation \(dS_{t}^0=rS_{t}^0dt,\;S_{0}^0=1,\) where \(r\) is the riskless interest rate. The remaining \(n\) asset prices \(S_{t}^{i}, i=1,\ldots,n\), evolve according to the linear stochastic differential equation \(dS_{t}^{i}=S_{t}^{i} [b_{t}^{i}+\sum_{j=1}^{d}\sigma_{ij}(t) dW_{t}^{j}]\). The main results are as follows. For the European exchange option \(X=(S_{t}^{m}-S_{t}^{m+1})^{+}\) with constant volatility matrix \((\sigma_{ij}), i,j=1,\ldots,d\), the price \(\pi\) associated with \(X\) is \[ \pi=S_{0}^{m}\Phi\left({\ln{S_{0}^{m}\over S_{0}^{m+1}}+{1\over 2}\sigma^2T\over\sigma\sqrt{T}}\right)-S_{0}^{m+1}\Phi\left({\ln{S_{0}^{m}\over S_{0}^{m+1}}-{1\over 2}\sigma^2T\over\sigma\sqrt{T}}\right), \] where \(\sigma=\sqrt{\sum_{k=1}^{d}(\sigma_{mk}-\sigma_{(m+1)k})^2}\) and \(\Phi(x)\) is the standard normal cumulative distribution function. For the European reverse exchange option \(Y=(S_{t}^{m+1}-S_{t}^{m})^{+}\) the price \(\pi_1\) is \(\pi_1=\pi-S_{0}^{m}+S_{0}^{m+1}\). The decomposition of Snell envelope and value function of the American exchange option is obtained.